**3. General principles of the new mechanism operating in hard rocks at high σ<sup>3</sup>**

#### **3.1 Structure of shear ruptures**

**Figure 4** illustrates the nature of shear rupture propagation in brittle intact rocks at high σ3. In **Figure 4a** a shear rupture propagates from left to right under stresses σ1, σ3, σn and τ representing the applied major and minor stresses and the induced normal and shear stresses. Shear ruptures are known to propagate through brittle rocks because of the creation of an echelon of tensile cracks at the rupture tip generated along the major stress that is at angle α<sup>o</sup> ≈ (30° ÷ 40°) to the shear rupture plane [25–28]. The echelon of inclined tensile cracks and inter-crack slabs forms a typical structure of shear ruptures illustrated by a photograph in **Figure 4b** (modified from [29]). Horizontal lines here indicate the rupture faces. It was observed

that at relative displacement of the rupture faces, inter-crack slabs are subject to rotation [25–28]. We will call hereafter the inter-crack slabs as domino blocks.

**Figure 4c** and **d** show two fundamentally different behaviours of domino blocks at rotation which characterise the conventional and the new understanding of failure mechanisms operating in hard rocks at high σ3. According to the conventional understanding in **Figure 4c**, domino blocks collapse at rotation creating friction within the process zone in the rupture head [25–28]. This mechanism is associated with frictional shear and provides conventional post-peak properties illustrated in **Figure 3b**. According to the new understanding (**Figure 4d**), domino blocks can withstand the rotation without collapse at a certain combination of such parameters as domino block geometry (ratio between the block length r and width w), rock strength (determining the strength of domino blocks) and applied stresses. Because the relative shear displacement of the rupture faces increases with distance from the rupture tip, the successively generated and rotated domino blocks form a fan-like structure that represents the rupture head [16–23]. This mechanism is associated with fan-hinged shear within the fan zone and with the following two fantastic features of the fan structure: (1) extremely low shear resistance approaching zero and (2) high amplification of shear stresses.

The fan mechanism is responsible for the 'abnormal' post-peak properties of rock specimens illustrated in **Figure 3c**. **Figure 4e** shows different stages of the fan-structure formation and propagation on the basis of tensile cracking process in a rock specimen. The orientation of tensile cracks and domino blocks in the propagating rupture tip is along σ1. However, due to relative displacement (shear) of the rupture faces, the domino blocks behind the tip consistently rotate and form the fan structure. Next sections considering unique features of the fan structure will demonstrate that after completion of the fan structure and up to the moment at which the fan has crossed the specimen body, the specimen strength is very low and corresponds to stages 1–4 in **Figure 3c**.

#### **3.2 Physical model of the fan mechanism**

The problem is that the direct experimental study of the fan mechanism is impossible today, firstly, because modern stiff and servocontrolled testing machines do not allow stopping the failure process governed by the fan mechanism and, secondly, because the fan structure is a transient phenomenon. The fan structure can be in the stable condition during its initial formation before the instability starts and at the final stage of the rupture termination. However,

**Figure 5.** *a) Transient nature of the fan structure and b) the conventional domino-like structure of shear ruptures.*

**67**

**Figure 6.**

*Illustration of the physical model of the fan structure.*

*Dramatic Weakening and Embrittlement of Intact Hard Rocks in the Earth's Crust at Seismic…*

in both cases during total unloading (e.g. associated with the removal of stresses from the specimen or with the tectonic exhumation of the rock mass involving the fan structure), reverse elastic deformations transform the fan into the conventional domino-block structure. **Figure 5a** illustrates this situation schematically on the laboratory specimen. The left specimen shows the fan structure formed during loading. The right specimen demonstrates that during unloading all domino blocks of the fan structure rotate backwards and finally form the conventional domino-like structure that can be seen in natural faults formed in intact

Due to the fact that direct experimental studies of the fan-structure properties are impossible, we will analyse them using a physical model. A video demonstrating the fan-structure formation and propagation along the model can be viewed at [30]. **Figure 6a** shows images of the physical model reflecting different stages of the fan-structure formation and propagation. **Figure 6a-I** shows the initial structure of the future fault that is represented by 'predetermined' domino blocks inclined at an angle α0 to the rupture plane. All blocks made from tiles are glued together to simulate a 'monolithic' material. The bond strength is less than that of the block material. The row of domino blocks is located between two layers, AB and CD, representing the two opposite faces of the rupture (elastic connectors). The upper and lower faces are fixed to the corresponding ends of each domino block. The

entire row of blocks is loaded with an evenly distributed weight, σp.

quent rotation of these blocks. We will consider this angle γ ≈ 800

that can cause the fan structure to propagate along the model.

between the upper AB and lower BC faces corresponds to angle γ = 400

For the physical model, the easiest way to apply shear stress τ uniformly distributed along the entire model is the inclination of the model by an angle γ. The distributed weight, σp in this case, creates a shear stress *τ* = *σp*sin(*γ*) and normal stress *σn* = *σp*cos(*γ*) along the model. By changing the angle γ, we can vary the applied stresses. Experiments conducted on the physical model show that at angle γ ≈ 800

upper face AB can move relative to the lower face CD due to the simultaneous separation from each other (tearing off) of all glued together domino blocks with the subse-

the material strength. At the absence of the domino structure, the frictional strength

However, if we form the fan structure from the domino blocks involved in the model, the upper face AB can be moved against the lower face CD at very low angles *γ* indicating very low shear resistance of the fan structure. Horizontal lines in **Figure 6b** indicate symbolically different levels of shear stresses: τs corresponds to the material strength, τf corresponds to the frictional strength, τfan corresponds to the fan-structure strength and τ corresponds to a low level of applied shear stress

It should be emphasised that for the initial formation of the fan structure, an additional local shear stress should be applied. In the physical model, the initial fan

, the

as corresponding to

.

*DOI: http://dx.doi.org/10.5772/intechopen.85413*

rocks (see photographs in **Figure 5b** [29]).

#### *Dramatic Weakening and Embrittlement of Intact Hard Rocks in the Earth's Crust at Seismic… DOI: http://dx.doi.org/10.5772/intechopen.85413*

in both cases during total unloading (e.g. associated with the removal of stresses from the specimen or with the tectonic exhumation of the rock mass involving the fan structure), reverse elastic deformations transform the fan into the conventional domino-block structure. **Figure 5a** illustrates this situation schematically on the laboratory specimen. The left specimen shows the fan structure formed during loading. The right specimen demonstrates that during unloading all domino blocks of the fan structure rotate backwards and finally form the conventional domino-like structure that can be seen in natural faults formed in intact rocks (see photographs in **Figure 5b** [29]).

Due to the fact that direct experimental studies of the fan-structure properties are impossible, we will analyse them using a physical model. A video demonstrating the fan-structure formation and propagation along the model can be viewed at [30]. **Figure 6a** shows images of the physical model reflecting different stages of the fan-structure formation and propagation. **Figure 6a-I** shows the initial structure of the future fault that is represented by 'predetermined' domino blocks inclined at an angle α0 to the rupture plane. All blocks made from tiles are glued together to simulate a 'monolithic' material. The bond strength is less than that of the block material. The row of domino blocks is located between two layers, AB and CD, representing the two opposite faces of the rupture (elastic connectors). The upper and lower faces are fixed to the corresponding ends of each domino block. The entire row of blocks is loaded with an evenly distributed weight, σp.

For the physical model, the easiest way to apply shear stress τ uniformly distributed along the entire model is the inclination of the model by an angle γ. The distributed weight, σp in this case, creates a shear stress *τ* = *σp*sin(*γ*) and normal stress *σn* = *σp*cos(*γ*) along the model. By changing the angle γ, we can vary the applied stresses. Experiments conducted on the physical model show that at angle γ ≈ 800 , the upper face AB can move relative to the lower face CD due to the simultaneous separation from each other (tearing off) of all glued together domino blocks with the subsequent rotation of these blocks. We will consider this angle γ ≈ 800 as corresponding to the material strength. At the absence of the domino structure, the frictional strength between the upper AB and lower BC faces corresponds to angle γ = 400 .

However, if we form the fan structure from the domino blocks involved in the model, the upper face AB can be moved against the lower face CD at very low angles *γ* indicating very low shear resistance of the fan structure. Horizontal lines in **Figure 6b** indicate symbolically different levels of shear stresses: τs corresponds to the material strength, τf corresponds to the frictional strength, τfan corresponds to the fan-structure strength and τ corresponds to a low level of applied shear stress that can cause the fan structure to propagate along the model.

It should be emphasised that for the initial formation of the fan structure, an additional local shear stress should be applied. In the physical model, the initial fan

**Figure 6.** *Illustration of the physical model of the fan structure.*

*Earth Crust*

that at relative displacement of the rupture faces, inter-crack slabs are subject to rotation [25–28]. We will call hereafter the inter-crack slabs as domino blocks.

and (2) high amplification of shear stresses.

corresponds to stages 1–4 in **Figure 3c**.

**3.2 Physical model of the fan mechanism**

**Figure 4c** and **d** show two fundamentally different behaviours of domino blocks at rotation which characterise the conventional and the new understanding of failure mechanisms operating in hard rocks at high σ3. According to the conventional understanding in **Figure 4c**, domino blocks collapse at rotation creating friction within the process zone in the rupture head [25–28]. This mechanism is associated with frictional shear and provides conventional post-peak properties illustrated in **Figure 3b**. According to the new understanding (**Figure 4d**), domino blocks can withstand the rotation without collapse at a certain combination of such parameters as domino block geometry (ratio between the block length r and width w), rock strength (determining the strength of domino blocks) and applied stresses. Because the relative shear displacement of the rupture faces increases with distance from the rupture tip, the successively generated and rotated domino blocks form a fan-like structure that represents the rupture head [16–23]. This mechanism is associated with fan-hinged shear within the fan zone and with the following two fantastic features of the fan structure: (1) extremely low shear resistance approaching zero

The fan mechanism is responsible for the 'abnormal' post-peak properties of rock specimens illustrated in **Figure 3c**. **Figure 4e** shows different stages of the fan-structure formation and propagation on the basis of tensile cracking process in a rock specimen. The orientation of tensile cracks and domino blocks in the propagating rupture tip is along σ1. However, due to relative displacement (shear) of the rupture faces, the domino blocks behind the tip consistently rotate and form the fan structure. Next sections considering unique features of the fan structure will demonstrate that after completion of the fan structure and up to the moment at which the fan has crossed the specimen body, the specimen strength is very low and

The problem is that the direct experimental study of the fan mechanism is impossible today, firstly, because modern stiff and servocontrolled testing machines do not allow stopping the failure process governed by the fan mechanism and, secondly, because the fan structure is a transient phenomenon. The fan structure can be in the stable condition during its initial formation before the instability starts and at the final stage of the rupture termination. However,

*a) Transient nature of the fan structure and b) the conventional domino-like structure of shear ruptures.*

**66**

**Figure 5.**

structure is generated by the application of force F to the leftmost domino block. When the local stress applied reaches a level τa, the front block will be torn off from the intact row, indicating the start of the tensile cracking process. The applied force is transmitted to the following blocks by elastically stretching the top rupture face (elastic connector), thereby causing the consecutive separation (tearing off) of the blocks and their rotation against the rupture faces. The fan formation is completed when the first block rotates to a total angle βtot = 180° − 2α0 at a shear displacement Δ of the upper face. The red graph MM in **Figure 6b** reflects the experimentally determined variation in shear resistance of the developing fan structure during its formation and further propagation of the completed fan. The rising resistance up to τs is associated with the formation of the first half of the fan, while the decreasing resistance corresponds to the second half formation. The reason for such variation in shear resistance will be discussed in the next section.

Experiments on the physical model show that the minimum angle γ at which the fan propagates spontaneously along the model is about 4°. Because the frictional strength for this model is characterised by γ ≈ 40°, we can conclude that shear resistance of the fan structure τfan is by the factor of ten less than the frictional strength: τfan ≈ 0.1τf. It should be emphasised that the low shear resistance takes place within the zone of the moving fan head only. In front of the fan, the material is in an intact condition (strength τs). Behind the fan shear resistance is equal to friction (strength τf). The fact that the fan structure can propagate through the intact model (representing the 'intact' material) at very low shear stresses applied indicates that the material strength in this case is determined by the shear resistance of the fan structure. For the physical model, the fan structure decreases the material strength by the ratio τs/τfan ≈ 14. We can suppose that for the rock specimen in **Figure 2d**, the fan mechanism decreases the material strength at the post-peak failure by the ratio ∆σs/∆σtr ≈ 30. The part of the post-peak curve in **Figure 3c** corresponding to the fan propagation through the intact rock specimen is represented by the horizontal line between stages 1 and 4.

#### **Figure 7.**

*Schematic explanation of the reasons for the low shear resistance of the fan structure. a) and b) Principle of self-balancing of the fan structure. c) Friction in joints of domino blocks. d) Distribution of shear resistance along the fault involving the fan structure.*

**69**

angle αo = 40o

the fan structure by Eq. (3):

For the ratio w/r = 0.1, we will have

*Dramatic Weakening and Embrittlement of Intact Hard Rocks in the Earth's Crust at Seismic…*

In order to cause the spontaneous rupture propagation through intact rock, the fan structure should provide, in addition to low shear resistance of the rupture head, also high shear stresses in the rupture tip and sufficiently high driving power at very low shear stresses applied. The next section explains unique principles

**3.3 Low shear resistance, high stress amplification and driving power generated** 

First, we will analyse the reason for the low shear resistance of the fan structure. Domino blocks in the fan are interconnected by the rupture faces and behave as hinges between the moving (sliding) rupture faces. Shear resistance of the fan structure τfan represents the resistance to displacement of the rupture faces relative to each other in the rupture head. **Figure 7** explains schematically the main principle responsible for low shear resistance of the fan structure. **Figure 7a** shows that all domino blocks in the fan are loaded by elementary forces N representing the normal stress σn. Elementary forces N applied to domino blocks in the front part of the fan resist to rotation of them. At the same time, elementary forces N applied to domino blocks in the rear part of the fan assist to rotation of these blocks. The key feature of the fan structure is the fact that each domino block in the front part of the fan is balanced by a symmetrical block of the rear part of the fan (**Figure 7b**). This means that the resistance to shear of this structure even at very high levels of normal stress

**Figure 7c** allows estimating roughly friction in joints. It shows a self-balancing domino block (representing the right block in **Figure 7b**) of length r and width w with cylindrical ends rotating with sliding friction in corresponding cylindrical grooves. The block is inclined at an angle α and loaded by force N. An analysis conducted in [20] shows that the presence of domino blocks between the shearing rupture faces decreases the resistance to shear compared with the conventional

τfan(α) = τ<sup>f</sup> w/r 2 sin<sup>2</sup>α (2)

as shown in **Figure 7d**. According to Eq. (2), the largest shear resis-

Eq. (2) shows that the resistance to shear τfan(α) between the rupture faces separated by self-balancing domino blocks and loaded by normal force N is determined by the conventional sliding friction τf, the ratio w/r and the angle α of the block inclination. Using Eq. (2) we can estimate the character of distribution of shear resistance between two rupture faces along the fan structure consisting of domino blocks characterised by the ratio w/r = 0.1 and by the initial (and final)

tance τfan(α) = 0.12 τf is provided by the front and the rear domino blocks of the fan. The minimum shear resistance τfan(α) = 0.05 τf is at the middle of the fan. The dotted curve in **Figure 7d** shows the distribution of shear resistance along the fan structure. In front of the fan, shear resistance is determined by the material strength τs, and behind the fan it corresponds to the frictional strength τf. Due to a small variation in shear resistance in the fan zone, we can describe the average strength of

τfan ≈ τ<sup>f</sup> w/r (3)

τfan ≈ 0.1τ<sup>f</sup> (4)

*DOI: http://dx.doi.org/10.5772/intechopen.85413*

**by the fan mechanism**

involved by the fan mechanism to satisfy these requirements.

applied will be determined solely by friction in joints.

frictional sliding in accordance with Eq. (2):

*Dramatic Weakening and Embrittlement of Intact Hard Rocks in the Earth's Crust at Seismic… DOI: http://dx.doi.org/10.5772/intechopen.85413*

In order to cause the spontaneous rupture propagation through intact rock, the fan structure should provide, in addition to low shear resistance of the rupture head, also high shear stresses in the rupture tip and sufficiently high driving power at very low shear stresses applied. The next section explains unique principles involved by the fan mechanism to satisfy these requirements.

## **3.3 Low shear resistance, high stress amplification and driving power generated by the fan mechanism**

First, we will analyse the reason for the low shear resistance of the fan structure. Domino blocks in the fan are interconnected by the rupture faces and behave as hinges between the moving (sliding) rupture faces. Shear resistance of the fan structure τfan represents the resistance to displacement of the rupture faces relative to each other in the rupture head. **Figure 7** explains schematically the main principle responsible for low shear resistance of the fan structure. **Figure 7a** shows that all domino blocks in the fan are loaded by elementary forces N representing the normal stress σn. Elementary forces N applied to domino blocks in the front part of the fan resist to rotation of them. At the same time, elementary forces N applied to domino blocks in the rear part of the fan assist to rotation of these blocks. The key feature of the fan structure is the fact that each domino block in the front part of the fan is balanced by a symmetrical block of the rear part of the fan (**Figure 7b**). This means that the resistance to shear of this structure even at very high levels of normal stress applied will be determined solely by friction in joints.

**Figure 7c** allows estimating roughly friction in joints. It shows a self-balancing domino block (representing the right block in **Figure 7b**) of length r and width w with cylindrical ends rotating with sliding friction in corresponding cylindrical grooves. The block is inclined at an angle α and loaded by force N. An analysis conducted in [20] shows that the presence of domino blocks between the shearing rupture faces decreases the resistance to shear compared with the conventional frictional sliding in accordance with Eq. (2):

$$
\pi\_{\text{fan}(\text{at})} = \pi\_{\text{f}} \le \text{w/r } 2 \sin^2 \alpha \tag{2}
$$

Eq. (2) shows that the resistance to shear τfan(α) between the rupture faces separated by self-balancing domino blocks and loaded by normal force N is determined by the conventional sliding friction τf, the ratio w/r and the angle α of the block inclination. Using Eq. (2) we can estimate the character of distribution of shear resistance between two rupture faces along the fan structure consisting of domino blocks characterised by the ratio w/r = 0.1 and by the initial (and final) angle αo = 40o as shown in **Figure 7d**. According to Eq. (2), the largest shear resistance τfan(α) = 0.12 τf is provided by the front and the rear domino blocks of the fan. The minimum shear resistance τfan(α) = 0.05 τf is at the middle of the fan. The dotted curve in **Figure 7d** shows the distribution of shear resistance along the fan structure. In front of the fan, shear resistance is determined by the material strength τs, and behind the fan it corresponds to the frictional strength τf. Due to a small variation in shear resistance in the fan zone, we can describe the average strength of the fan structure by Eq. (3):

$$
\boldsymbol{\pi}\_{\text{fan}} \approx \boldsymbol{\pi}\_{\text{f}} \,\mathbf{w}/\mathbf{r} \tag{3}
$$

For the ratio w/r = 0.1, we will have

$$
\tau\_{\rm fan} \approx \mathbf{0.1} \tau\_{\rm f} \tag{4}
$$

*Earth Crust*

structure is generated by the application of force F to the leftmost domino block. When the local stress applied reaches a level τa, the front block will be torn off from the intact row, indicating the start of the tensile cracking process. The applied force is transmitted to the following blocks by elastically stretching the top rupture face (elastic connector), thereby causing the consecutive separation (tearing off) of the blocks and their rotation against the rupture faces. The fan formation is completed when the first block rotates to a total angle βtot = 180° − 2α0 at a shear displacement Δ of the upper face. The red graph MM in **Figure 6b** reflects the experimentally determined variation in shear resistance of the developing fan structure during its formation and further propagation of the completed fan. The rising resistance up to τs is associated with the formation of the first half of the fan, while the decreasing resistance corresponds to the second half formation. The reason for such variation

Experiments on the physical model show that the minimum angle γ at which the fan propagates spontaneously along the model is about 4°. Because the frictional strength for this model is characterised by γ ≈ 40°, we can conclude that shear resistance of the fan structure τfan is by the factor of ten less than the frictional strength: τfan ≈ 0.1τf. It should be emphasised that the low shear resistance takes place within the zone of the moving fan head only. In front of the fan, the material is in an intact condition (strength τs). Behind the fan shear resistance is equal to friction (strength τf). The fact that the fan structure can propagate through the intact model (representing the 'intact' material) at very low shear stresses applied indicates that the material strength in this case is determined by the shear resistance of the fan structure. For the physical model, the fan structure decreases the material strength by the ratio τs/τfan ≈ 14. We can suppose that for the rock specimen in **Figure 2d**, the fan mechanism decreases the material strength at the post-peak failure by the ratio ∆σs/∆σtr ≈ 30. The part of the post-peak curve in **Figure 3c** corresponding to the fan propagation through the intact rock specimen is represented by the horizontal

*Schematic explanation of the reasons for the low shear resistance of the fan structure. a) and b) Principle of self-balancing of the fan structure. c) Friction in joints of domino blocks. d) Distribution of shear resistance* 

in shear resistance will be discussed in the next section.

line between stages 1 and 4.

**68**

**Figure 7.**

*along the fault involving the fan structure.*

#### **Figure 8.** *Principles of shear stress amplification by the fan mechanism.*

#### **Figure 9.**

*Relative distribution of shear resistance and amplified shear stresses in the rupture head caused by the fan mechanism.*

This estimation is in accord with experimental results obtained on the physical model.

**Figure 8** illustrates schematically the unknown before and very powerful principle of shear stress amplification inherent in the fan mechanism. It shows the fan structure propagating from left to right through the intact material under the effect of distributed shear stress τ0 applied to the whole fault. Behind the fan, domino blocks completed their rotation and form the frictional zone. The horizontal line τ0 on the graph above the fan indicates the level of applied shear stress that is significantly less than the material strength τs. The explanation on how the fan mechanism can provide high stresses equal to τs in the rupture tip at very low shear stresses applied τ0 is as follows.

All domino blocks of the fan structure are loaded by elementary forces fτ representing the applied shear stress. Within the fan zone, domino blocks are separated. Due to this, elementary forces fτ applied to each block cause corresponding rotation of them and stretch the elastic connector (i.e. the upper rupture face) in front

**71**

**Figure 10.**

*Dramatic Weakening and Embrittlement of Intact Hard Rocks in the Earth's Crust at Seismic…*

of each block, thus transmitting these forces to the rupture tip. This principle is similar to one shown below the fan. According to this principle, real forces affecting domino blocks within the fan zone increase towards the front as shown on the graph above the fan structure (line NN). This means that shear stress within the fan zone increases from the level τ1 at the rear end of the fan towards the rupture tip approximately in proportion to the number of domino blocks k involved in the fan structure. We will call this stress as the fan-transient stress: τtr ~ τ1k. Analysis conducted in [18–20] shows that shear ruptures propagating in real materials can involve the fan structure with about a thousand of domino blocks. This means that the potential ability of the fan mechanism in stress amplification can be very large,

**Figure 9** illustrates the relation between shear resistance and stresses generated by the fan mechanism at low shear stresses applied. A shaded curve here reflects the distribution of shear resistance along a fault involving the fan structure. In front of the fan, shear resistance corresponds to the material strength τs, behind the fan it is determined by friction τf and in the fan zone shear resistance is significantly less than the frictional strength τfan <sup>≪</sup> τf. A solid curve reflects the shear stress variation. This schema shows that the fan mechanism can provide the rupture development at very low shear stresses applied τ0 which can be significantly below the frictional strength, i.e. at τfan < τ<sup>0</sup> <sup>≪</sup> τf. The coloured area here represents symbolically the power generated by the fan mechanism that causes spontaneous rupture development even at low shear stresses applied. The instability is resulted from the fact that in the fan zone the generated stress exceeds the shear resistance and in the rupture tip it corresponds to the material strength.

It should be emphasised that despite the fact that the shear rupture here propagates through 'intact' material, the magnitude of stress drop Δτ = τo - τ1 can be very low (lower than at the frictional stick-slip instability) because the rupture propagates at low shear stresses applied τ<sup>0</sup> <sup>≪</sup> τf. Depending on the level of τ0, the fan mechanism can provide two types of rupture mode (crack-like and pulselike) observed in natural and laboratory earthquakes [4, 31, 32]. This question

*The evolution of failure mechanisms in hard rocks with rising confining pressure σ3 and variable efficiency of* 

*the fan mechanism within the pressure range σ3fan(min) < σ3 < σ3fan(max).*

but the real generated stresses are limited by the material strength.

*DOI: http://dx.doi.org/10.5772/intechopen.85413*

#### *Dramatic Weakening and Embrittlement of Intact Hard Rocks in the Earth's Crust at Seismic… DOI: http://dx.doi.org/10.5772/intechopen.85413*

of each block, thus transmitting these forces to the rupture tip. This principle is similar to one shown below the fan. According to this principle, real forces affecting domino blocks within the fan zone increase towards the front as shown on the graph above the fan structure (line NN). This means that shear stress within the fan zone increases from the level τ1 at the rear end of the fan towards the rupture tip approximately in proportion to the number of domino blocks k involved in the fan structure. We will call this stress as the fan-transient stress: τtr ~ τ1k. Analysis conducted in [18–20] shows that shear ruptures propagating in real materials can involve the fan structure with about a thousand of domino blocks. This means that the potential ability of the fan mechanism in stress amplification can be very large, but the real generated stresses are limited by the material strength.

**Figure 9** illustrates the relation between shear resistance and stresses generated by the fan mechanism at low shear stresses applied. A shaded curve here reflects the distribution of shear resistance along a fault involving the fan structure. In front of the fan, shear resistance corresponds to the material strength τs, behind the fan it is determined by friction τf and in the fan zone shear resistance is significantly less than the frictional strength τfan <sup>≪</sup> τf. A solid curve reflects the shear stress variation. This schema shows that the fan mechanism can provide the rupture development at very low shear stresses applied τ0 which can be significantly below the frictional strength, i.e. at τfan < τ<sup>0</sup> <sup>≪</sup> τf. The coloured area here represents symbolically the power generated by the fan mechanism that causes spontaneous rupture development even at low shear stresses applied. The instability is resulted from the fact that in the fan zone the generated stress exceeds the shear resistance and in the rupture tip it corresponds to the material strength.

It should be emphasised that despite the fact that the shear rupture here propagates through 'intact' material, the magnitude of stress drop Δτ = τo - τ1 can be very low (lower than at the frictional stick-slip instability) because the rupture propagates at low shear stresses applied τ<sup>0</sup> <sup>≪</sup> τf. Depending on the level of τ0, the fan mechanism can provide two types of rupture mode (crack-like and pulselike) observed in natural and laboratory earthquakes [4, 31, 32]. This question

#### **Figure 10.**

*The evolution of failure mechanisms in hard rocks with rising confining pressure σ3 and variable efficiency of the fan mechanism within the pressure range σ3fan(min) < σ3 < σ3fan(max).*

*Earth Crust*

**70**

model.

**Figure 9.**

**Figure 8.**

*Principles of shear stress amplification by the fan mechanism.*

*mechanism.*

stresses applied τ0 is as follows.

This estimation is in accord with experimental results obtained on the physical

All domino blocks of the fan structure are loaded by elementary forces fτ representing the applied shear stress. Within the fan zone, domino blocks are separated. Due to this, elementary forces fτ applied to each block cause corresponding rotation of them and stretch the elastic connector (i.e. the upper rupture face) in front

**Figure 8** illustrates schematically the unknown before and very powerful principle of shear stress amplification inherent in the fan mechanism. It shows the fan structure propagating from left to right through the intact material under the effect of distributed shear stress τ0 applied to the whole fault. Behind the fan, domino blocks completed their rotation and form the frictional zone. The horizontal line τ0 on the graph above the fan indicates the level of applied shear stress that is significantly less than the material strength τs. The explanation on how the fan mechanism can provide high stresses equal to τs in the rupture tip at very low shear

*Relative distribution of shear resistance and amplified shear stresses in the rupture head caused by the fan* 

is discussed in [19, 20]. The fan mechanism explains also the heat flow paradox observed for extreme ruptures [6, 13].
